{
  "abstract": "We develop an efficient $C^1$ Bernstein quasi-Trefftz discretization for heterogeneous high-frequency Helmholtz problems on unstructured triangular meshes. The method first imposes strong $C^1$ continuity through Bernstein--Bézier smoothness relations on local macro-patches and then reduces the resulting conforming polynomial space by enforcing projected Helmholtz residual constraints. On two-triangle patches, this compresses the local dimension from quadratic growth in the polynomial degree to a trace-sized space of dimension $2p+1$ under the natural rank condition. Variable matrix-valued coefficients are handled through local polynomial projection, allowing coefficient-approximation effects to be separated from discretization error. A graph-residual formulation couples the reduced patch spaces and produces a sparse global system in compressed coordinates. The implementation combines explicit $C^1$ continuation, batched coefficient projection, batched residual construction, and stable local kernel extraction by QR factorization. Numerical experiments on heterogeneous unstructured meshes demonstrate high-order accuracy, roundoff-level $C^1$ conformity, robust high-frequency resolution, and substantial reductions in local and global computational cost. Turning-point and penetrable-scattering examples further illustrate the flexibility of the approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05849",
  "author_total": 1,
  "authors": [
    {
      "name": "Shelvean Kapita"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fb2dcxnp66enk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05849",
  "fulltext_url": null,
  "id": "fb2dcxnp66enk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05849",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7211988397",
  "pdf_url": "https://arxiv.org/pdf/2609.05849",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fb2dcxnp66enk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05849",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05849"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Efficient $C^1$ Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems",
  "title_text": "Efficient C^1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fb2dcxnp66enk",
  "venue": "arXiv",
  "versions": []
}